Relative Yield Total

In addition to conducting our analysis with biomass as the measurement of ecosystem function, in this document we report our results using Relative Yield Total (Loreau and Hector 2001; Wagg et al. 2019).

Mirroring the manuscript’s central analysis, our models for the across-treatment effect were encoded as: RelativeYieldTotal ~ -1 + Stage + Stage:Shannon, and the within-treatment effect was encoded as: RelativeYieldTotal ~ -1 + Richness:Stage + Richness:Stage:Shannon. All models successfully converged, with Rhat values of 1.0, and posterior predictive checks (PPC) were used to visually validate the model fits.

1 Figure 2 - Counter-gradient

The relationship between Shannon diversity and Relative Yield Total was qualitatively similar to that of Shannon diversity for the majority of the models (5/6).

Within Grass3, positive within-treatment slopes estimating total biomass flip to negative when estimating Relative Yield Total. This leads the model to more closely reflect the other two grassland models. Because conspecific negative density dependence (CNDD) is present within the model, species’ monoculture biomasses are dramatically reduced in comparison to their biomasses within communities. Communities made of more competitive species will have lower Shannon diversity and higher Relative Yield Total, because there are few species present but they are released from the negative influence of CNDD. High diversity communities will generally have less total biomass per species, making each species’ contribution to Relative Yield Total smaller. Because the positive influence of CNDD saturates as individuals become surrounded by heterospecifics, this decrease in Relative Yield Total is not outweighed by any corresponding increases in the strength of CNDD. This creates negative within-treatment relationships between Shannon diversity and Relative Yield Total.

2 Figure 3 - Across-treatment effect

Considering the relationship between our measure of the internal coexistence processes within each model and the across-treatment effect of realized diversity on Relative Yield Total, we find that the aggregate patterns are nearly identical to those of total biomass.

3 Figure 4 - Within-treatment effect

Considering the relationship between our measure of the internal coexistence processes within each model and the within-treatment effect of realized diversity on Relative Yield Total, we find that the aggregate patterns are nearly identical to those of total biomass.

4 Model validation

This section of the document describes the statistical models’ validation, using Shannon diversity as the focal biodiversity metric and Relative Yield Total as the focal ecosystem function.

Important terms:

4.1 Grass1

Clark, A. T., C. Lehman, and D. Tilman. 2018. Identifying mechanisms that structure ecological communities by snapping model parameters to empirically observed trade-offs. Ecology Letters 21:494–505.

4.1.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 642) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain              -14.82      5.37   -25.37    -4.21 1.00     2003     1860
## StageWithoutseedrain           -27.46      5.76   -38.81   -16.00 1.00     1718     1844
## StageWithseedrain:Shannon       17.32      2.08    13.30    21.33 1.00     2038     1924
## StageWithoutseedrain:Shannon    26.56      2.60    21.46    31.66 1.00     1712     2074
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    22.58      0.65    21.36    23.89 1.00     2624     2106
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error     Q2.5     Q97.5
## R2 0.2128202 0.02514011 0.163788 0.2614227

4.1.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.1.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.1.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.1.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.1.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                25.80     22.30   -17.44    69.12 1.01     6006     3054
## Ninitial4:StageWithseedrain                93.73     20.16    54.85   133.85 1.00     4772     2716
## Ninitial8:StageWithseedrain               127.34     16.23    95.09   158.47 1.00     5285     2956
## Ninitial16:StageWithseedrain              216.62     14.21   188.05   244.77 1.00     6024     2579
## Ninitial32:StageWithseedrain              250.36     16.71   216.95   283.24 1.00     5654     2640
## Ninitial2:StageWithoutseedrain              3.83      8.61   -13.21    20.40 1.00     5441     2667
## Ninitial4:StageWithoutseedrain             38.47     12.33    14.02    61.99 1.00     5499     2860
## Ninitial8:StageWithoutseedrain             66.81     12.34    42.85    90.99 1.00     5254     3246
## Ninitial16:StageWithoutseedrain           163.64     15.55   133.39   193.91 1.00     5492     2757
## Ninitial32:StageWithoutseedrain           174.27     19.85   135.11   212.55 1.00     5103     2850
## Ninitial2:StageWithseedrain:Shannon       -12.15     13.88   -39.15    14.78 1.01     5984     3101
## Ninitial4:StageWithseedrain:Shannon       -36.26      9.23   -54.71   -18.44 1.00     4808     2652
## Ninitial8:StageWithseedrain:Shannon       -39.54      6.13   -51.40   -27.44 1.00     5328     2977
## Ninitial16:StageWithseedrain:Shannon      -59.82      4.82   -69.35   -50.19 1.00     5963     2604
## Ninitial32:StageWithseedrain:Shannon      -61.76      5.27   -72.15   -51.12 1.00     5585     2587
## Ninitial2:StageWithoutseedrain:Shannon      0.99      5.79   -10.07    12.52 1.00     5587     2762
## Ninitial4:StageWithoutseedrain:Shannon    -12.78      6.32   -25.05    -0.24 1.00     5471     2913
## Ninitial8:StageWithoutseedrain:Shannon    -19.85      5.46   -30.58    -9.30 1.00     5282     3039
## Ninitial16:StageWithoutseedrain:Shannon   -49.75      6.41   -62.10   -37.26 1.00     5541     2939
## Ninitial32:StageWithoutseedrain:Shannon   -43.65      7.57   -58.18   -28.76 1.00     5060     2962
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    12.72      0.36    12.02    13.42 1.00     7473     2989
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate   Est.Error      Q2.5     Q97.5
## R2 0.7499603 0.008676023 0.7318249 0.7657269

4.1.2.1 Posterior predictive checks

4.1.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.1.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.1.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.2 Grass2

Turnbull, L. A., J. M. Levine, M. Loreau, and A. Hector. 2013. Coexistence, niches and biodiversity effects on ecosystem functioning. Ecology Letters 16:116–127.

4.2.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 642) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain                3.11      1.33     0.54     5.77 1.00     1770     1782
## StageWithoutseedrain            -5.47      1.43    -8.26    -2.66 1.00     1861     1908
## StageWithseedrain:Shannon        6.26      0.43     5.39     7.11 1.00     1782     1857
## StageWithoutseedrain:Shannon    12.65      0.58    11.50    13.80 1.00     1816     1834
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     6.95      0.20     6.58     7.35 1.00     2804     2405
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error     Q2.5     Q97.5
## R2 0.5269179 0.01839451 0.489196 0.5598625

4.2.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.2.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.2.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.2.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.2.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain               178.46     19.56   139.60   217.41 1.00     5531     2739
## Ninitial4:StageWithseedrain               218.83     20.20   178.90   259.29 1.00     5266     2855
## Ninitial8:StageWithseedrain               186.97     18.12   152.57   221.97 1.00     4432     2768
## Ninitial16:StageWithseedrain              138.56     28.38    83.44   194.73 1.00     4827     2868
## Ninitial32:StageWithseedrain              102.51     34.09    37.72   171.06 1.00     5255     2911
## Ninitial2:StageWithoutseedrain             23.47     12.90    -2.24    49.25 1.00     4773     2937
## Ninitial4:StageWithoutseedrain            -24.26      7.60   -39.35    -9.37 1.00     4865     2737
## Ninitial8:StageWithoutseedrain            -14.80      6.04   -26.96    -2.93 1.00     4361     2615
## Ninitial16:StageWithoutseedrain             9.26      6.20    -2.91    21.34 1.00     5185     2860
## Ninitial32:StageWithoutseedrain            36.63      8.10    20.18    52.49 1.00     5072     2708
## Ninitial2:StageWithseedrain:Shannon      -101.24     11.85  -125.05   -77.65 1.00     5525     2778
## Ninitial4:StageWithseedrain:Shannon       -87.77      8.76  -105.25   -70.44 1.00     5270     2922
## Ninitial8:StageWithseedrain:Shannon       -56.06      6.25   -68.10   -44.01 1.00     4450     2735
## Ninitial16:StageWithseedrain:Shannon      -31.73      8.00   -47.66   -16.13 1.01     4816     2867
## Ninitial32:StageWithseedrain:Shannon      -17.71      8.13   -34.05    -2.34 1.00     5270     2867
## Ninitial2:StageWithoutseedrain:Shannon     -7.70      7.97   -23.57     8.22 1.00     4813     2825
## Ninitial4:StageWithoutseedrain:Shannon     22.44      4.13    14.42    30.50 1.00     4861     2575
## Ninitial8:StageWithoutseedrain:Shannon     19.15      2.74    13.79    24.62 1.00     4352     2636
## Ninitial16:StageWithoutseedrain:Shannon     8.01      2.27     3.54    12.50 1.00     5160     2757
## Ninitial32:StageWithoutseedrain:Shannon    -0.60      2.42    -5.31     4.40 1.00     5078     2728
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     5.10      0.14     4.82     5.39 1.00     6722     3015
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate   Est.Error      Q2.5     Q97.5
## R2 0.7489654 0.008953145 0.7303685 0.7649965

4.2.2.1 Posterior predictive checks

4.2.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.2.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.2.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.3 Grass3

May, F., V. Grimm, and F. Jeltsch. 2009. Reversed effects of grazing on plant diversity: The role of below-ground competition and size symmetry. Oikos 118:1830–1843.

4.3.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 642) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain              -10.03      0.83   -11.71    -8.45 1.00     2181     1807
## StageWithoutseedrain           -10.05      0.83   -11.71    -8.43 1.00     1899     1651
## StageWithseedrain:Shannon        4.10      0.29     3.56     4.68 1.00     2205     1657
## StageWithoutseedrain:Shannon     4.32      0.30     3.72     4.95 1.00     1841     1837
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     3.85      0.11     3.63     4.06 1.00     2806     2038
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.3965623 0.02392655 0.3482008 0.4415797

4.3.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.3.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.3.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.3.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.3.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                37.34     17.42     3.83    70.87 1.00     3770     3156
## Ninitial4:StageWithseedrain                75.41     11.98    51.92    99.07 1.00     4000     3132
## Ninitial8:StageWithseedrain                57.71     11.92    34.59    81.15 1.00     4076     3284
## Ninitial16:StageWithseedrain               56.38     10.70    34.86    77.25 1.00     4055     3157
## Ninitial32:StageWithseedrain               42.56     14.14    15.37    69.38 1.00     3978     2998
## Ninitial2:StageWithoutseedrain             -4.76      3.04   -10.85     1.01 1.00     3612     2514
## Ninitial4:StageWithoutseedrain             19.80      5.14     9.46    29.73 1.00     3863     3057
## Ninitial8:StageWithoutseedrain             20.36      5.41     9.55    30.76 1.00     4331     3068
## Ninitial16:StageWithoutseedrain            50.84     10.07    31.14    70.67 1.00     4388     2891
## Ninitial32:StageWithoutseedrain            58.53     14.92    29.42    88.31 1.00     3617     2640
## Ninitial2:StageWithseedrain:Shannon       -24.33     10.44   -44.38    -4.27 1.00     3752     3113
## Ninitial4:StageWithseedrain:Shannon       -33.38      5.25   -43.76   -23.06 1.00     3997     3122
## Ninitial8:StageWithseedrain:Shannon       -19.97      4.21   -28.21   -11.77 1.00     4065     3313
## Ninitial16:StageWithseedrain:Shannon      -15.75      3.17   -21.96    -9.40 1.00     4037     3112
## Ninitial32:StageWithseedrain:Shannon       -9.46      3.71   -16.51    -2.39 1.00     3987     3057
## Ninitial2:StageWithoutseedrain:Shannon      0.84      1.88    -2.80     4.59 1.00     3606     2512
## Ninitial4:StageWithoutseedrain:Shannon     -9.69      2.40   -14.32    -4.86 1.00     3878     3023
## Ninitial8:StageWithoutseedrain:Shannon     -7.39      2.06   -11.32    -3.30 1.00     4348     3035
## Ninitial16:StageWithoutseedrain:Shannon   -15.37      3.16   -21.62    -9.17 1.00     4397     2889
## Ninitial32:StageWithoutseedrain:Shannon   -14.56      4.24   -23.09    -6.34 1.00     3642     2373
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     3.14      0.09     2.98     3.33 1.00     7302     2604
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.6015697 0.01540604 0.5709656 0.6298231

4.3.2.1 Posterior predictive checks

4.3.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.3.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.3.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.4 Forest1

Rüger, N., R. Condit, D. H. Dent, S. J. DeWalt, S. P. Hubbell, J. W. Lichstein, O. R. Lopez, C. Wirth, and C. E. Farrior. 2020. Demographic trade-offs predict tropical forest dynamics. Science 368:165–168.

4.4.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 642) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain               13.11      2.75     7.65    18.34 1.00     1991     1991
## StageWithoutseedrain            15.61      2.92    10.01    21.37 1.00     1902     2193
## StageWithseedrain:Shannon        7.66      1.55     4.73    10.74 1.00     1978     1905
## StageWithoutseedrain:Shannon     6.31      2.04     2.20    10.28 1.00     1940     2212
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    15.33      0.43    14.49    16.19 1.00     2852     2361
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate  Est.Error       Q2.5      Q97.5
## R2 0.05731091 0.01686307 0.02755178 0.09281329

4.4.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.4.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.4.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.4.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.4.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                44.73      7.27    30.32    58.80 1.00     4528     2782
## Ninitial4:StageWithseedrain                61.57      6.11    49.94    73.80 1.00     4655     2536
## Ninitial8:StageWithseedrain                66.74      7.00    53.04    80.60 1.00     4205     3151
## Ninitial16:StageWithseedrain               56.76      8.82    39.32    74.04 1.00     4858     3188
## Ninitial32:StageWithseedrain               28.88     13.56     2.10    55.62 1.00     4324     2842
## Ninitial2:StageWithoutseedrain             31.55      9.25    13.32    49.28 1.00     3814     2994
## Ninitial4:StageWithoutseedrain             48.19      5.69    37.36    59.66 1.00     3869     3238
## Ninitial8:StageWithoutseedrain             69.04      6.04    57.21    80.98 1.00     4878     2736
## Ninitial16:StageWithoutseedrain            32.04      6.12    20.00    43.66 1.00     4728     2810
## Ninitial32:StageWithoutseedrain            20.55      5.56     9.84    31.43 1.00     4202     3149
## Ninitial2:StageWithseedrain:Shannon       -29.62      6.05   -41.36   -17.57 1.00     4526     2679
## Ninitial4:StageWithseedrain:Shannon       -31.45      4.58   -40.56   -22.59 1.00     4558     2524
## Ninitial8:StageWithseedrain:Shannon       -23.97      4.49   -32.64   -15.08 1.00     4080     3202
## Ninitial16:StageWithseedrain:Shannon      -11.71      4.42   -20.30    -3.04 1.00     4886     3024
## Ninitial32:StageWithseedrain:Shannon        2.80      5.49    -8.06    13.74 1.00     4345     2935
## Ninitial2:StageWithoutseedrain:Shannon    -21.18      8.68   -37.90    -3.91 1.00     3925     3032
## Ninitial4:StageWithoutseedrain:Shannon    -24.93      4.63   -34.38   -16.04 1.00     3905     3079
## Ninitial8:StageWithoutseedrain:Shannon    -30.72      4.55   -39.67   -21.62 1.00     4822     2657
## Ninitial16:StageWithoutseedrain:Shannon    -0.21      3.95    -7.82     7.57 1.00     4784     2806
## Ninitial32:StageWithoutseedrain:Shannon     6.73      3.09     0.73    12.73 1.00     4330     2846
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    11.25      0.32    10.64    11.90 1.00     8368     2653
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5    Q97.5
## R2 0.5039633 0.01964115 0.4633785 0.539916

4.4.2.1 Posterior predictive checks

4.4.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.4.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.4.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.5 Forest2

Maréchaux, I., and J. Chave. 2017. An individual-based forest model to jointly simulate carbon and tree diversity in Amazonia: description and applications. Ecological Monographs.

4.5.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 642) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain                1.61      1.03    -0.48     3.56 1.00     1941     1867
## StageWithoutseedrain             0.31      1.24    -2.15     2.72 1.00     1809     1985
## StageWithseedrain:Shannon       -0.21      0.36    -0.90     0.50 1.00     1899     1888
## StageWithoutseedrain:Shannon    -4.80      0.64    -6.01    -3.55 1.00     1835     2070
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     5.84      0.16     5.54     6.16 1.00     2878     2639
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.4437236 0.02170308 0.3990911 0.4846449

4.5.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.5.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.5.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.5.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.5.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                17.29      6.82     3.75    30.33 1.00     3913     2796
## Ninitial4:StageWithseedrain                46.38      7.95    30.97    62.16 1.00     5153     3161
## Ninitial8:StageWithseedrain                29.79     11.68     7.22    52.35 1.00     3392     2711
## Ninitial16:StageWithseedrain               45.52     19.25     8.08    82.69 1.00     3526     3086
## Ninitial32:StageWithseedrain               83.10     27.42    29.91   136.66 1.00     3620     2871
## Ninitial2:StageWithoutseedrain             -3.77      3.68   -10.85     3.40 1.00     3881     2776
## Ninitial4:StageWithoutseedrain            -10.70      3.53   -17.48    -3.65 1.00     3874     2990
## Ninitial8:StageWithoutseedrain             -6.03      3.87   -13.54     1.63 1.00     3901     2896
## Ninitial16:StageWithoutseedrain           -14.19      3.33   -20.69    -7.70 1.00     4292     2898
## Ninitial32:StageWithoutseedrain           -13.61      4.84   -23.14    -4.15 1.00     4599     2960
## Ninitial2:StageWithseedrain:Shannon       -11.24      4.48   -19.87    -2.38 1.00     3944     2839
## Ninitial4:StageWithseedrain:Shannon       -21.64      3.76   -29.06   -14.35 1.00     5213     3188
## Ninitial8:StageWithseedrain:Shannon       -10.27      4.36   -18.63    -1.88 1.00     3311     2774
## Ninitial16:StageWithseedrain:Shannon      -13.38      5.75   -24.48    -2.07 1.00     3520     3192
## Ninitial32:StageWithseedrain:Shannon      -20.87      6.97   -34.50    -7.30 1.00     3613     2916
## Ninitial2:StageWithoutseedrain:Shannon      0.36      2.72    -5.03     5.65 1.00     3840     2805
## Ninitial4:StageWithoutseedrain:Shannon      2.57      2.22    -1.81     6.84 1.00     3891     2992
## Ninitial8:StageWithoutseedrain:Shannon     -1.54      1.93    -5.33     2.27 1.00     3928     2789
## Ninitial16:StageWithoutseedrain:Shannon     1.11      1.57    -1.89     4.14 1.00     4219     2858
## Ninitial32:StageWithoutseedrain:Shannon     0.33      1.95    -3.52     4.15 1.00     4557     3218
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     5.36      0.15     5.08     5.66 1.00     7919     3241
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.5408732 0.01758599 0.5049973 0.5747371

4.5.2.1 Posterior predictive checks

4.5.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.5.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.5.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.6 Dryland

Reineking, B., M. Veste, C. Wissel, and A. Huth. 2006. Environmental variability and allocation trade-offs maintain species diversity in a process-based model of succulent plant communities. Ecological Modelling.

4.6.1 Across-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Stage + Stage:Shannon 
##    Data: d_ (Number of observations: 642) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                              Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## StageWithseedrain                0.20      1.93    -3.49     4.08 1.00     2168     2104
## StageWithoutseedrain            -4.07      2.38    -8.75     0.60 1.01     2192     2206
## StageWithseedrain:Shannon        0.79      0.76    -0.68     2.23 1.00     2118     2105
## StageWithoutseedrain:Shannon    -0.19      1.42    -2.99     2.60 1.00     2167     2107
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma    10.07      0.27     9.57    10.61 1.00     2663     2675
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##      Estimate  Est.Error       Q2.5     Q97.5
## R2 0.09981672 0.02094118 0.06084377 0.1425852

4.6.1.1 Posterior predictive checks

We next use posterior predictive checks (PPC) to judge the fit of the model. These compare the real data to the posterior distribution, conditioned on the observed data.

4.6.1.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.6.1.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y). A line indicates a 1:1 correspondence for reference.

4.6.1.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.

4.6.2 Within-treatment effect

A summary table of the BRMS model results:

##  Family: gaussian 
##   Links: mu = identity; sigma = identity 
## Formula: relative_yield_total ~ -1 + Ninitial:Stage + Ninitial:Stage:Shannon 
##    Data: d_ (Number of observations: 640) 
## Samples: 4 chains, each with iter = 2000; warmup = 1000; thin = 1;
##          total post-warmup samples = 4000
## 
## Population-Level Effects: 
##                                         Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## Ninitial2:StageWithseedrain                -5.07      7.67   -20.16     9.85 1.00     3809     3031
## Ninitial4:StageWithseedrain               -15.28     10.52   -36.05     5.10 1.00     3659     3120
## Ninitial8:StageWithseedrain               -34.97     12.91   -60.10    -8.67 1.00     3770     2949
## Ninitial16:StageWithseedrain              -34.01     18.29   -69.55     2.04 1.00     3269     2941
## Ninitial32:StageWithseedrain              -72.47     23.74  -116.97   -25.30 1.00     3158     2920
## Ninitial2:StageWithoutseedrain             -2.04      5.07   -11.66     8.01 1.00     3978     2893
## Ninitial4:StageWithoutseedrain            -20.20      5.23   -30.26    -9.54 1.00     4210     2693
## Ninitial8:StageWithoutseedrain            -21.93      5.41   -32.33   -11.26 1.00     4112     2804
## Ninitial16:StageWithoutseedrain           -36.46      7.57   -51.36   -21.61 1.00     3949     2974
## Ninitial32:StageWithoutseedrain           -44.48     11.38   -67.33   -22.32 1.00     3581     2963
## Ninitial2:StageWithseedrain:Shannon         5.30      5.27    -5.08    15.57 1.00     3885     3061
## Ninitial4:StageWithseedrain:Shannon         9.10      5.50    -1.46    19.88 1.00     3671     2979
## Ninitial8:StageWithseedrain:Shannon        15.27      5.32     4.58    25.81 1.00     3812     2848
## Ninitial16:StageWithseedrain:Shannon       12.10      6.19     0.05    24.20 1.00     3303     2937
## Ninitial32:StageWithseedrain:Shannon       21.65      6.85     8.11    34.60 1.00     3075     2932
## Ninitial2:StageWithoutseedrain:Shannon      3.27      4.03    -4.48    11.28 1.00     4024     2669
## Ninitial4:StageWithoutseedrain:Shannon     12.43      3.57     5.20    19.33 1.00     4156     2969
## Ninitial8:StageWithoutseedrain:Shannon     10.14      3.23     3.77    16.43 1.00     4152     2907
## Ninitial16:StageWithoutseedrain:Shannon    15.78      4.05     7.89    23.70 1.00     3922     3122
## Ninitial32:StageWithoutseedrain:Shannon    18.05      5.68     7.15    29.30 1.00     3609     3091
## 
## Family Specific Parameters: 
##       Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
## sigma     9.33      0.27     8.82     9.86 1.00     7696     3081
## 
## Samples were drawn using sampling(NUTS). For each parameter, Bulk_ESS
## and Tail_ESS are effective sample size measures, and Rhat is the potential
## scale reduction factor on split chains (at convergence, Rhat = 1).

Note the Rhat summary column: variation from 1.0 indicates the the model did not converge.

The Bayesian R-squared:

##     Estimate  Est.Error      Q2.5     Q97.5
## R2 0.2566906 0.02466921 0.2071976 0.3037057

4.6.2.1 Posterior predictive checks

4.6.2.1.1 Density plot

The density of both the real data (y, black line), and from fitted draws of the models (y_rep, blue lines).

4.6.2.1.2 Scatter plot

Average prediction (y_rep) for each real data point (y), grouping the comparison of y to y_rep by Ninitial. A line indicates a 1:1 correspondence for reference.

4.6.2.1.3 Highest-density interval

Highest-density interval (HDI) for each effect within the model. This characterizes the uncertainty of our posterior distributions. Highest-density intervals can be thought of as credibility intervals (see here). We use the 89% HDI as recommended by Kruschke (2014), see here for more information.